6 problems

For $n\ge10$, let $P(n)$ be the set of partitions of $n$, and let $[\lambda]$ denote the irreducible character of $S_n$ indexed by $\lambda$. Refinement of the HSTZ conjecture. The…

Let $n\in\mathbb N$ and let $\lambda\vdash n$. A character is multiplicity-free when every irreducible constituent occurs with multiplicity at most one. The partitions listed in pa…

Let $n=\tfrac{1}{2}k(k+1)$ and let $\rho_k=(k,\ldots,2,1)\vdash n$ be the staircase partition. Write $S^2([\rho_k])$ and $A^2([\rho_k])$ for the symmetric and alternating parts of…

Let $\Lambda_n$ be the ring of homogeneous symmetric functions of degree $n$ with Kronecker product $*$, and define … where $s_\lambda$ is the Schur function indexed by $\lambda$.…

Let $d$ be a positive integer, let $\ol\nu\vdash d$, and let $s_{\ol\nu}(x)$ be the polynomial map associated with the staircase Kronecker-square coefficients. Let $c(\ol\nu)$ be t…

Let $d$ be a positive integer, let $\ol\nu\vdash d$, and let $s_{\ol\nu}$ be the polynomial map associated with the staircase Kronecker-square coefficients. Let $c(\ol\nu)$ and…